We develop a Bayesian methodology for numerical solution of the incompressible Navier--Stokes equations with quantified uncertainty. The central idea is to treat discretized Navier--Stokes dynamics as a state-space model and to view numerical solution as posterior computation: priors encode physical structure and modeling error, and the solver outputs a distribution over states and quantities of interest rather than a single trajectory. In two dimensions, stochastic representations (Feynman--Kac and stochastic characteristics for linear advection--diffusion with prescribed drift) motivate Monte Carlo solvers and provide intuition for uncertainty propagation. In three dimensions, we formulate stochastic Navier--Stokes models and describe particle-based and ensemble-based Bayesian workflows for uncertainty propagation in spectral discretizations. A key computational advantage is that parameter learning can be performed stably via particle learning: marginalization and resample--propagate (one-step smoothing) constructions avoid the weight-collapse that plagues naive sequential importance sampling on static parameters. When partial observations are available, the same machinery supports sequential observational updating as an additional capability. We also discuss non-Gaussian (heavy-tailed) error models based on normal variance-mean mixtures, which yield conditionally Gaussian updates via latent scale augmentation.

翻译：我们提出了一种用于求解不可压缩 Navier-Stokes 方程并量化其不确定性的贝叶斯方法。其核心思想是将离散化的 Navier-Stokes 动力学视为一个状态空间模型，并将数值求解视为后验计算：先验编码物理结构和建模误差，求解器输出的是关于状态和感兴趣量的分布，而非单一轨迹。在二维情况下，随机表示（针对具有指定漂移项的线性对流-扩散方程的 Feynman-Kac 公式和随机特征线法）启发了蒙特卡洛求解器，并为不确定性传播提供了直观理解。在三维情况下，我们构建了随机 Navier-Stokes 模型，并描述了基于粒子和基于集合的贝叶斯工作流，用于谱离散化中的不确定性传播。一个关键的计算优势在于，可以通过粒子学习稳定地进行参数学习：边缘化以及重采样-传播（一步平滑）构造避免了在静态参数上进行朴素序贯重要性采样时常见的权重退化问题。当可获得部分观测数据时，同一套机制支持作为附加功能的序贯观测更新。我们还讨论了基于正态方差-均值混合的非高斯（重尾）误差模型，该模型通过潜在尺度增强产生条件高斯更新。